Secondary structure kinetics model

5.2 Appendix to Chapter 2

5.2.4 Secondary structure kinetics model

Multistrand [153] essentially employs a Gillespie algorithm [73] for generating statistically correct trajectories of a stochastic Markov process. Code implementing the Multistrand model is available for public download (http://www.dna.caltech.edu/Multistrand/).

Choice and timing of next transition. Suppose the box is in statei. Then, the next statemis chosen randomly from the statesjwhich are adjacent toi(i.e. they differ fromiby only one base pair), weighted by the rate of transition to each.

P(statemis chosen) = kim

P

jkij

(5.54)

The time taken for the transition fromitomto occur (∆t) is chosen randomly from an exponential distribution with rate parameterλ, whereλis the total rate of transitioning from the current state i. That is,

P(time taken is∆t) =λe^−λ∆t (5.55)

whereλ=P

jk_ij.

First step mode.We will describe first step mode for a general reaction of the form:

A+B−−→^keff C+D (5.56)

For a strand displacement reaction, which involves intermediate steps, this model assumes a low- concentration regime where the bimolecular step occurs on a much longer timescale than the uni-

molecular steps; equation5.56may then be used to accurately capture the dynamics.

The first step simulation mode begins with the bimolecular join step where A and B collide and form a base pair. The secondary structures of A and B are obtained by Boltzmann sampling the non-pseudoknotted secondary structure space for each molecule. If the bimolecular reaction rates are slow enough for the reactants to be in equilibrium, this sampling is valid. Once the secondary structures of A and B have been sampled, one of the available join steps is chosen at random and the simulation proceeds. As more trajectories are run, different initial secondary structures for A and B and different join steps are explored.

Note that we are not directly simulating the bimolecular join steps, whose rates are propor- tional to the simulated concentration (and are hence much slower than the unimolecular steps).

This allows Multistrand to focus on the trajectories where a collision does occur, rather than spend- ing most of the time simulating unimolecular reactions while waiting for the rare bimolecular reaction.

As simulation of a trajectory proceeds, two distinct end states are tracked: the molecules falling apart into the reactants (one of the A + B configurations), or forming the products (one of the C + D configurations). Each trajectory simulated may be classified as one that failed (if the former happens) or one that reacted (if the latter happens).

Our simulations yield the following data: first passage times for trajectories that reacted (∆tⁱ_react), first passage times for trajectories that failed (∆tⁱ_fail), the number of trajectories that reacted (N_react) and failed (N_fail), and the estimated average rate of collision (k_collin /M/s) of the reactants A and B. For each trajectory, the rate of collision is calculated asN_first×k_bi×u, whereN_firstis the number of possible first base pairs between the sampled secondary structures of A and B for that trajectory anduis the simulated concentration. k_collis estimated as the mean of the rates of collision for the trajectories simulated.

The following model is used to analyze simulation data. We assume that molecules A and B collide to either form a reactive molecule that will yield the products C and D (in our case, successful displacement) or a nonreactive molecule that will fall apart into the reactants A and B in some time (in our case, unsuccessful displacement).

A+B−→^k1 AB−→^k2 C+D (5.57)

A + B ^k

0

−−*1

)−−

k₂⁰

AB⁰ (5.58)

Our model (equations5.57,5.58) is fitted as follows.

k1= N_react

N_react+N_fail ×k_coll (5.59)

k⁰₁= N_fail

N_react+N_fail ×k_coll (5.60)

k₂= 1

E[∆tⁱ_react] (5.61)

k⁰₂= 1

E[∆tⁱ_fail] (5.62)

Assuming equation5.56is valid,k_effmay be predicted [153] based on our model as follows.

k_eff= 1

∆t_correct ×1

u (5.63)

where∆t_correct is the expected time taken for a successful reaction to occur. ∆t_correct is calculated from the expected time for a failed collision to fall apart into the reactants (∆t_fail), and the expected time for a reactive collision to produce the products (∆t_react). ∆t_fail and ∆t_react depend on the expected time for any collision to occur (∆t_coll). These quantities are calculated as follows.

∆t_correct = ∆t_fail×k⁰₁

k₁ + ∆t_react (5.64)

∆t_fail= ∆t_coll+ 1

k₂⁰ (5.65)

∆t_react= ∆t_coll+ 1

k₂ (5.66)

∆t_coll = 1

(k1+k⁰₁)×u (5.67)

In the low-concentration regime, the resolution of the three-stranded complex (resulting in successful displacement of the incumbent or dissociation of the invader) may be assumed to be effectively instantaneous compared to the rate of the bimolecular collision step. That is, we may assumek_coll×u << k2, k⁰₂. Indeed, we make this assumption since we are inferring a bimolecular rate constant (equation5.56). With that assumption, the general formulation (equation5.63) may be reduced to

k_eff=k_coll×p (5.68)

wherepis the probability that the collision results in successful displacement of the incumbent.

Simulation details. We simulated the “average strength toehold” experimental system of Zhang and Winfree [147], measuring strand displacement rates as a function of toehold length

0 5 10 15 0

2 4 6 8

Toehold length log 10(k eff)

Dangles = Some Dangles = None Dangles = All Matching toehold

 

 

Experiment Experiment

A B

0 5 10 15

0 2 4 6 8

Toehold length

Metropolis Kawasaki

 

 

Figure 5.6: Multistrand simulations at 25^◦C with different choices: (A) (i) in treating free energy con- tributions due to dangles [178] (options “Some”(default), “None” and “All” in the NUPACK [123] energy model [120]) and (ii) with substrate overhangs only as long as the toehold on the invader - i.e. matching length bottom toeholds on the substrate, rather than the full 15 base overhang used by Zhang and Winfree [147] (B) different ways of assigning absolute transition rates for unimolecular steps while satisfying detailed balance. Note that none of the variations are able to account for the experimental data points (in black) from Zhang and Winfree [147]; solid black line is their phenomenological model. Standard errors for Multistrand simulations are under 1% (not shown).

Strand Sequence

Substrate 5⁰-GAAGTGACATGGAGACGTAGGGTATTGAATGAGGG -3⁰ Incumbent 5⁰- CCCTCATTCAATACCCTACG -3⁰

Invader 5⁰- CCCTCATTCAATACCCTACGTCTCCATGTCACTTC-3⁰

Table 5.3: Sequences used in Multistrand simulations of strand displacement, with toeholds in italics. For toehold lengths less than 15, the toehold of the invader is truncated to the appropriate length, measured from the 5’ end. For simulations with a matching length substrate overhang, the toehold of the substrate was also truncated to match the toehold of the invader.

at 25^◦C. We omitted the downstream step used for experimental detection purposes, and deleted the extra domain in the incumbent which was used only in that step. The sequences we used are provided in Table5.3. Simulations were performed in first step mode.

Multistrand variations. The experimental system of Zhang and Winfree [147] employs a sub- strate strand with a 15-base overhang. Depending on the length of the invading toehold, a subset of this overhang is complementary to the toehold. The fact that the substrate overhang is longer than the toehold it binds to could conceivably have two effects: (i) stabilizing the first toehold base pair between the invader and the substrate through a dangle free energy contribution and/or (ii) allowing unexpected pathways of displacement through a larger set of possible first base pairs.

Multistrand simulations with a matching length substrate overhang (truncated to match the length of the invading toehold) are closer to experiment by only 0.6 orders of magnitude (Figure5.6(A)).

A B

ï 0 4 6 0

4 6 8

log₁₀(k_uni/k_bi) log 10(k eff)

Toehold: 15 Toehold: 0

1 0 10

2 3 4

log₁₀(k_uni/k_bi)

Orders of mag. acceleration Multistrand

IEL

Figure 5.7: The dependence of Multistrand predictions onkuni/k_bi(red circles indicate default value of k_uni/k_bi). Error bars are3∗SE long where SE is the standard error. (A) Orders of magnitude acceleration in k_effbetween toehold lengths 15 and 0 (A15,0), as predicted by Multistrand, as a function oflog₁₀(kuni/k_bi).

IEL(2.6, 0) predictions are shown for comparison. Even implausibly low values ofk_uni/k_bido not result in acceleration that matches the experimentally observed value of 6.5 orders of magnitude. (B)log₁₀(keff)vs log₁₀(k_uni/k_bi)for toehold lengths 15 and 0. The surprising non-monotonicity in (A) at the lowest value of kuni/kbiis observed to arise from the disproportionately large decrease inlog₁₀(keff)for toehold length 15 in (B). We hypothesize that this is likely a sequence dependent effect.

This suggests that possible effects (i) and (ii) are not large enough to explain the discrepancy be- tween Multistrand predictions and experimental measurements of strand displacement kinetics.

Because the energy models used by Multistrand [153], NUPACK [123], Mfold [184], and Vienna RNA [183] do not have a consensus method for handling dangle energy terms, we ran Multistrand simulations with each of the three options (Figure5.6(A)). For each dangles option,k_biandk_uni were separately calibrated to the same data (hybridization, zippering) used for calibrating the Metropolis method, as described in Schaeffer [153]. Only minor differences were observed.

We also performed Multistrand simulations using the Kawasaki [174] method for assigning unimolecular transition rates, for which k_bi and k_uni were also recalibrated; again this yielded nearly identical results (Figure5.6(B)).

Dependence on the ratiok_uni/k_bi.When the invader is bound to the substrate by just one base of the toehold, it can either dissociate, leading to unsuccessful displacement, or form another base pair of the toehold and proceed towards zippering. Since dissociation is a bimolecular process, its rate is influenced byk_bi, while the rate of the unimolecular zippering process is influenced byk_uni. Decreasingk_uni/k_biincreases the rate of the former relative to the latter. Although this is true for both short and long toeholds, short toeholds reach the state where the invader is bound to the sub-

strate by just one base more often than long toeholds. So, decreasingk_uni/k_bidisproportionately reduces the displacement rate of short toeholds and increases the orders of magnitude acceleration due to toehold length predicted by Multistrand (Figure5.7).